Appendix 1 – Terms and Concepts in Model Estimationagrino.org/nchiras/research/pubs/thesis/Appendices.pdfAppendix 1 – Terms and Concepts in Model Estimation 311 The variances of

308

Appendix 1 – Terms and Concepts in Model Estimation

Any quantities calculated from measurements disturbed by noise will only be estimates

of the unknown parameters of the system being modelled. The quality of a given

estimate can be assessed by its:

• Bias – the systematic error on the value obtained

• Variance - a measure of the uncertainty on the estimate.

The first- and second-order moments of a random variable x are termed its mean and

variance and defined as

}{)( xEx =µ (1A-1)

}}){{()( 22 xExEx −=σ (1A-2)

where E{•} denotes the expected value. The standard deviation is the square root of the

variance and hence has the same units as the mean value. These quantities must

normally be estimated from a finite set of N samples.

∑=

=N

iix

Nx

1

1 (1A-3)

2

1

2 )(1

1∑=

−−

=N

ii xx

Ns (1A-4)

Appendix 1 – Terms and Concepts in Model Estimation 309

which are known as the sample means and the sample variance respectively. The

variance of the sample mean is given by s2/N. The sample variance of a complex

random variable z = x + jy is given by

*

1

2 ))((1

1zzzz

Ns i

N

ii −−

−= ∑

= (1A-5)

If the variable has a complex normal distribution then the real and imaginary parts are

independent and normally distributed, with equal variances. The complex variance is

then twice the variance on either the real or imaginary parts.

For a column vector variable x, the covariance matrix is defined as

])(|)[()( TEEC xxxxx −−= (1A-6)

where the element cij of C(x) is the covariance between the elements i and j of x. It can

also be expressed in terms of a “mean of the squares” matrix and the square of the

means

TTTTTT EEEEEEEC xxxxxxxxxxxxx −=⋅+−⋅−= ][ ][)( (1A-7)

There are a number of key definitions in the classifications of estimators:

Unbiased: an estimator is unbiased if the mathematical expectation of the estimates

equals the true parameter values, p0

0)ˆ( pp =E (1A-8)

Asymptotically unbiased: if the expected value converges to the true value as the

number of measurements N increases


0)ˆ(lim pp =∞→

EN

(1A-9)

where N denotes the number of data points, which can be interchangeably the number of

time-domain samples or the number of frequencies.

Consistent: an estimator is consistent if Np̂ converges in some sense to p0 as N

approaches infinity. Several convergence concepts have been defined, which

convergence:

--- in mean square 0}ˆ{lim 0 =−∞→

pp NN

E (1A-10)

--- with probability one 1}ˆ{lim 0 ==∞→

pp NN

E (1A-11)

--- in probability 0 0} ˆ {lim 0 >∀=>−∞→

δδpp NN

E (1A-12)

where the probability of the bias being greater than any nonzero value δ is zero N ��

Convergence in mean square implies that the estimator is asymptotically unbiased.

Convergence in mean square and convergence with probability 1 are affinal, but not

identical, properties. Convergence in mean square and probability one both imply

convergence in probability but not vice versa. An estimator which exhibits convergence

with probability one is termed strongly consistent, while an estimator which only

exhibits convergence in probability is termed weakly consistent.

Efficient: an unbiased estimator E1 is efficient if C1<C2 for all unbiased estimators E2,

where C1 and C2 are the parameter covariance matrices, which relate the parameter

values p0 and the estimated values Np̂

} ]ˆ][ˆ[ {)ˆ( 00 pppppC −−= NNN E (1A-13)


The variances of the individual elements of Np̂ make up the principal diagonal of

)ˆ( NpC . A desirable property of the estimator is that the variance, or uncertainty, of the

estimates is as small as possible. However, there exists a lower bound on the covariance

matrix of an unbiased estimator, termed the Cramer-Rao bound, which is expressed as

-1 )ˆ( FpC ≥N (1A-14)

where F is the Fisher information matrix, a measure of the amount of information

present, in relation to the parameters. It is possible to construct an estimator with a

covariance matrix lower than the Cramer-Rao bound but it will be biased. The

information matrix applies to any unbiased of the parameters p using the measurements

y

00|

)|p(yln |

)|p(yln

)|p(yln E

2

2

ppppp

pp

pp

p==

∂∂−=

∂

∂

∂

∂= EFT

(1A-15)

where p(y | p) is the joint probability density function of the output in relation to the

parameter vector and the matrix of second derivatives is known as the Hessian. If p is a

d-dimensional column vector then the first derivatives will also form a d-dimensional

column vector and the Hessian will be d × d matrix.

Normal: it is highly desirable that the parameter estimates are normally distributed,

since a normally distributed variable is completely defined by its first- and second-

order moments and this allows statistical bounds to be calculated in a straightforward

manner.

Robust: if some properties of the estimator are still valid when all the a priori

assumptions are met. Most commonly this relates to a violation of the assumptions


regarding the noise. If the estimator can still produce a consistent estimate then it will

converge to the true value as N ��

Maximum Likelihood Estimator (MLE): the MLE is a strongly consistent and

asymptotically unbiased estimator if the true system belongs to the estimator model set.

In the presence of independent, identically distributed (iid) measurements it is also

asymptotically efficient, since it approaches the Cramer-Rao lower bound

asymptotically. If there is an estimator which reaches this lower bound it will be an

MLE. The parameter estimates are also asymptotically normal.

The primary sources for the summary presented in this Appendix were:

Åström, K. J. (1980). “Maximum likelihood and prediction error methods”, Automatica,

vol. 16, pp. 551-574.

Norton, J.P. (1986). An Introduction to Identification, Academic Press, London.

Ljung, L. (1987). System Identification - Theory for the User, Prentice Hall, Englewood

Cliffs.

Appendix 2 – Linear Gas Turbine Modelling Results 313

Appendix 2 - Linear Gas Turbine Modelling Results

________________________________________________

Table 2A-1

SMALL-SIGNAL TESTS USED FOR LINEAR GAS TURBINE MODELLING

LEVEL/TEST SIGNAL AMP

(±%Wf)

Fmin

(Hz)

Fmax

(Hz)

Bit interval

(ms)

No. bits

Level-55% NH

Test 2A

Test 3A

Test 1C

Multisine (1)

IRMLBS

GP Multisine

10

10

10

0.005

0.005

0.010

0.355

0.549

1.050

---

800

---

---

7

---

Level-65% NH

Test 4A

Test 10B

IRMLBS

IRMLBS

10

10

0.005

0.010

0.549

0.110

800

400

7

7

Level-75% NH

Test 6A

Test 7A

Test 8A

Test 11B

Multisine (1)

IRMLBS

Multisine (2)

IRMLBS

10

10

10

10

0.005

0.005

0.005

0.010

0.355

0.549

0.555

0.110

---

800

---

400

---

7

---

7

Level-80% NH

Test 12C GP Multisine 10 0.010 1.050 --- ---

Level-85% NH

Test 13A

Test 12 B

IRMLBS

IRMLBS

10

10

0.005

0.010

0.549

0.110

800

400

7

7

Level-90% NH

Test 11A

Test 12A

Test 19C

Multisine (1)

IRMLBS

GP Multisine

10

10

10

0.005

0.005

0.010

0.355

0.549

1.050

---

800

---

---

7

---


0 20 40 60 80 100 120 140 160 180 200−1

−0.5

0

0.5

1

Time (s)

Am

p

(a)

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40

0.05

0.1

0.15

0.2

Frequency (Hz)

Am

p

(b)

Figure 2A-1. One period of Multisine (1) in (a) time domain and (b) frequency domain.

The time domain amplitude is normalised to one.

0 20 40 60 80 100 120 140 160 180 200−1

−0.5

0

0.5

1

Time (s)

Am

p

(a)

0 0.1 0.2 0.3 0.4 0.5 0.60

0.02

0.04

0.06

0.08

0.1

0.12

Frequency (Hz)

Am

p

(b)

Figure 2A-2. One period of Multisine (2) in (a) time domain and (b) frequency domain.



0 10 20 30 40 50 60 70 80 90 100−1

−0.5

0

0.5

1

Time (s)

Am

p

(a)

0.2 0.4 0.6 0.8 1 1.20

0.05

0.1

0.15

Frequency (Hz)

Am

p

(b)

Figure 2A-3. One period of GP Multisine in (a) time domain and (b) frequency domain.


0 20 40 60 80 100 120 140 160 180 200

−1

−0.5

0

0.5

1

Time (s)

Am

p

(a)

0.5 1 1.5 2 2.50

0.05

0.1

0.15

0.2

Frequency (Hz)

Am

p

(b)

f(−3db)

Figure 2A-4. One period of IRMLBS signal in (a) time domain and (b) frequency

domain.


Frequency Domain Estimation Results

Test at 55% NH

Test A3 HP Shaft

Order K Kmin DC gain Delay

(ms)

σ% Poles σ% Zeros σ%

0/1 482.1 54.5 0.274405 69.37 1.87 -0.2331 0.45 --- ---

1/2 69.24 53.5 0.262191 36.05 6.56 -0.8908

-0.2765

7.96

1.37

-1.0977 7.57

2/3 59.13 52.5 .026080 15.51 48.43 -0.4253

-3.4530

-0.3184

24.23

20.38

11.61

-0.5567

-4.2539

13.91

25.41

Test A3 LP Shaft


(ms)


0/1 476.2 54.5 0.202359 67.36 1.82 -0.2497 0.44 --- ---

1/2 97.61 53.5 0.194985 34.89 6.99 -1.0448

-0.2898

8.08

1.17

-1.1758 7.96

Test A2 HP Shaft


(ms)


0/1 96.1 18.5 0.26096 64.93 3.27 -0.2538 0.43 --- ---

1/2 37.09 17.5 0.257366 39.39 12.32 -0.8970

-0.2717

20.85

1.46

-0.9967 20.87

Test A2 LP Shaft


(ms)


0/1 110.9 18.5 0.195108 59.30 3.49 -0.2705 0.41 --- ---

1/2 55.77 17.5 0.192502 37.85 10.37 -0.8350

-0.2907

18.74

1.54

-0.9259 18.37


Test C1 HP Shaft


(ms)


0/1 263.0 25.5 0.252527 58.77 1.414 -0.250 0.47 --- ---

1/2 26.57 24.5 0.246696 36.17 5.72 -2.31

-0.269

9.907

0.743

-2.67 10.75

2/3 24.09 23.5 0.244586 34.27 6.95 +0.195

-2.480

-0.267

70.64

10.33

0.831

+0.193

-2.89

70.94

11.39

Test C1 LP Shaft


(ms)


0/1 84.68 25.5 0.171629 46.09 2.01 -0.285 0.48 --- ---

1/2 50.54 24.5 0.171155 17.07 57.9 -10.4

-0.289

23.4

0.53

-15.9 48.5

2/3 43.12 23.5 0.172292 31.28 10.9 -2.29+1.9i

-2.29-1.9I

-0.284

21.9

21.9

0.73

-2.49+1.87i

-2.49-1.87i

24.9

24.9

3/4 41.32 22.5 0.170802 29.1 13.44 +0.244

-2.45+1.6i

-2.45-1.6i

-0.284

77.5

22.1

22.1

0.82

+0.242

-2.71+1.48i

-2.71+1.48i

77.8

25.9

25.9

Test at 65% NH

Test A4 HP Shaft


(ms)


0/1 96.28 54.5 0.170403 19.17 8.65 -0.4134 0.30 --- ---

1/2 53.05 53.5 0.179336 21.27 8.11 -0.4226

-0.0485

0.50

25.78

-0.0518 25.07


Test A4 LP Shaft


(ms)


0/1 2147 54.5 0.134446 -23.78 2.79 -0.4840 0.14 --- ---

1/2 259.4 53.5 0.137963 7.587 13.93 -0.3675

-0.9669

1.12

2.74

-0.6961 3.51

2/3 85.46 52.5 0.14252 11.93 10.93 -0.0464

-1.3418

-0.4243

13.54

4.5

1.14

-0.0492

-1.0632

13.44

4.93

Test B10 HP Shaft


(ms)


0/1 90.69 54.5 0.174287 26.91 9.66 -0.394 0.539 --- ---

2/3 86.83 52.5 0.174292 27.03 10.31 -0.394

0.005+3.17i

0.005-3.17i

0.54

559.53

559.53

0.003+3.1i

0.003-3.1i

698

698

Test B10 HP Shaft (41 harmonics only BW=0.81Hz)Order K Kmin DC gain Delay

(ms)


0/1 67.42 39.5 0.174265 27.47 10.37 -0.394 0.546 --- ---

2/3 58.33 37.5 0.174424 23.47 13.33 -0.393

-0.081+1.8i

-0.081-1.8i

0.58

39.47

39.47

-0.08+1.8i

-0.08-1.8i

37.7

37.7

Test B10 LP Shaft (Outlier removed)Order K Kmin DC gain Delay

(ms)


0/1 423.9 54.5 0.133075 -27.06 38.01 -0.482 0.36 --- ---

1/2 115.6 53.5 0.138779 13.72 10.28 -1.26

-0.4

6.66

1.52

-1.02 7.41

2/3 64.6 52.5 0.137129 12.5 11.48 -0.400

+0.228

-1.25

1.54

36.4

6.67

+0.225

-1.01

36.6

7.46


Test at 75% NH

Test A6 HP Shaft


(ms)


0/1 739.4 18.5 0.122747 2.953 19.11 -0.5512 0.12 --- ---

1/2 29.66 17.5 0.126552 10.42 6.39 -0.2440

-0.6159

5.11

0.78

-0.2746 5.63

Test A6 LP Shaft


(ms)


0/1 3139 18.5 0.091591 -84.1 1.23 -0.6949 0.19 --- ---

1/2 92.08 17.5 0.095619 -4.655 44.59 -0.4461

-1.4982

1.22

2.20

-0.8080 2.69

2/3 23.25 16.5 0.097819 4.427 63.59 -1.001

-0.5252

-1.8185

17.66

2.38

3.95

-0.1061

-1.0704

17.99

4.67

Test A8 HP Shaft


(ms)


0/1 196.4 54.5 0.121233 10.72 4.38 -0.5522 0.18 --- ---

1/2 75.14 53.5 0.127907 12.22 4.11 -0.1388

-0.5742

13.9

0.51

-0.1514 13.7

Test A8 LP Shaft


(ms)


0/1 3263 54.5 0.087608 -41.26 1.80 -0.7692 0.18 --- ---

1/2 142.7 53.5 0.095151 11.49 11.51 -0.4929

-1.8592

1.07

2.09

-1.0698 2.48

2/3 88.53 52.5 0.098083 15.13 10.14 -0.0995

-0.5552

-2.1097

17.62

1.97

3.04

-0.1067

-1.2872

17.94

3.64


Test A7 HP Shaft


(ms)


0/1 265.2 54.5 0.122594 7.164 8.05 -0.5601 0.19 --- ---

1/2 99.87 53.5 0.127526 10.38 6.17 -0.1768

-0.5927

10.6

0.68

-0.1929 10.9

Test A7 LP Shaft


(ms)


0/1 4365 54.5 0.089021 -4.14 1.67 -0.7672 0.19 --- ---

1/2 272.0 53.5 0.095997 3.30 34.15 -0.4511

-1.5442

1.03

1.6

-0.8318 2.17

2/3 93.3 52.5 0.099271 9 15.45 -0.0351

-0.5113

-1.7837

2.39

1.38

2.39

-0.0372

-1.0383

13.76

3

Test B11 HP Shaft


(ms)


0/1 167.6 54.5 0.124708 4.64 11.59 -0.554 0.25 --- ---

1/2 72.47 53.5 0.130437 6.38 8.97 -0.241

-0.604

12.9

1.50

-0.272 13.8

2/3 70.07 52.5 0.1305 10.39 21.0 -10.6

-0.592

-0.216

29.8

1.43

14.5

-10.1

-0.240

15.3


Test B11 LP Shaft


(ms)


0/1 2906 54.5 0.086560 -17.96 3.17 -0.858 28.27 --- ---

1/2 152.7 53.5 0.096080 9.95 7.93 -1.98

-0.497

1.94

1.39

-1.1 2.70

2/3 136.2 52.5 0.098392 11.04 7.63 -2.15

-0.567

-0.162

2.82

3.72

25.79

-1.28

-0.175

4.37

26.99

Test at 80% NH

Test C12 HP Shaft


(ms)


0/1 253.9 25.5 0.108624 17.24 1.67 -0.6116 0.19 --- ---

1/2 250.8 24.5 0.10867 19.11 4.41 -0.6125

+10.38

23.22

0.201

+10.6 23.97

Test C12 LP Shaft


(ms)


0/1 4263 25.5 0.071771 -2.597 17.07 -1.0214 0.24 --- ---

1/2 95.36 24.5 0.080049 20.658 2.96 -0.4916

-1.9945

1.39

1.35

-0.9429 2.26


Test at 85% NH

Test A13 HP Shaft


(ms)


0/1 552.1 54.5 0.084858 -5.239 10.4 -0.7480 0.20 --- ---

1/2 87.03 53.5 0.088965 4.55 16.1 -0.4181

-0.9267

5.01

1.96

-0.528 6.56

2/3 82.25 52.5 0.089196 7.20 16.6 -2.13

-0.821

-0.336

16.48

3.06

9.18

-2.03

-0.391

16.86

10.96

Test A13 LP Shaft


(ms)


0/1 3555 54.5 0.060413 -49.35 1.49 -1.1817 0.25 --- ---

1/2 96.87 53.5 0.065905 4.852 28.9 -0.5575

-2.2616

1.64

1.64

-0.9428 2.50

2/3 47.34 52.5 0.067243 12.309 16.65 -0.2457

0.8003

-2.7452

13.0

5.76

4.06

-0.2787

-1.4092

14.72

6.65

Test B12 HP Shaft


(ms)


0/1 437.4 54.5 0.0784 -2.60 11.8 -0.8360 0.29 --- ---

1/2 106.8 53.5 0.0836 1.34 29.89 -0.5296

-1.1664

5.10

3.76

-0.773 8.30

2/3 92.67 52.5 0.085129 2.836 19.44 -2.34

-0.827

-0.254

12.8

3.26

16.8

-2.14

-0.289

13.9

18.1


Test B12 LP Shaft


(ms)


0/1 2549 54.5 0.055121 -1506 3.30 -1.494 0.31 --- ---

1/2 93.40 53.5 0.064147 6.449 11.16 -0.5716

-2.4590

2.23

1.35

-0.9726 2.89

2/3 79.79 52.5 0.065189 8.305 10.85 -0.3452

-0.8984

-2.7307

16.9

13.5

3.54

-0.4172

-1.4061

22.1

10.7

Test B12 LP Shaft (54 harmonics)


(ms)


0/1 2502 52.5 0.055210 -15.84 3.21 -1.49 0.31 --- ---

1/2 81.54 51.5 0.064143 6.486 11.57 -2.46

-0.572

1.38

2.23

-0.974 2.91

2/3 67.52 50.5 0.065068 8.790 11.2 -2.81

-1.00

-0.380

4.33

14.6

13.1

-1.53

-0.479

12.3

18.4

Test at 90% NH

Test A11 HP Shaft


(ms)


0/1 438.1 18.5 0.072370 -18.71 5.45 -0.806 0.24 --- ---

1/2 13.33 17.5 0.074789 -1.77 86.2 -0.3995

-0.9858

6.20

2.0

-0.4807 7.65


Test A11 LP Shaft


(ms)


0/1 2227 18.5 0.052836 -98.41 1.37 -1.1767 0.30 --- ---

1/2 52.68 17.5 0.056086 -27.259 10.04 -0.4360

-1.8773

2.77

1.78

-0.5908 3.52

2/3 28.46 16.5 0.056468 2.582 306.3 -0.3070

-1.0717

-3.0830

14.3

9.74

14.3

-0.3640

-1.7276

9.51

17.15

Test A12 HP Shaft


(ms)


0/1 803.3 54.5 0.071250 -6.13 6.59 -0.8487 0.16 --- ---

1/2 91.63 53.5 0.075407 1.6 35.34 -0.3990

-0.9973

4.95

1.16

-0.4838 5.72

2/3 78.07 52.5 0.075627 7.693 21.74 -4.29

-0.930

-0.328

19.4

1.42

6.91

-4.03

-0.380

18.4

7.61

Test A12 LP Shaft


(ms)


0/1 4212 54.5 0.050133 -40.27 1.52 -1.4466 0.23 --- ---

1/2 136.7 53.5 0.055793 1.56 70.74 -0.5612

-2.3458

1.85

1.19

-0.8470 2.37

2/3 98.69 52.5 0.056970 6.712 21.58 -0.2525

-0.8214

-2.6531

12.2

5.86

2.60

-0.2843

-1.2317

13.5

6.02


Test C19 HP Shaft


(ms)


0/1 297.5 25.5 0.073881 6.848 4.32 -0.8438 0.30 --- ---

1/2 40.06 24.5 0.077288 11.12 3.93 -0.5921

-1.2632

4.81

5.52

-0.9112 9.79

Test C19 LP Shaft


(ms)


0/1 3386 25.5 0.045415 -0.73 49.64 -1.7264 0.25 --- ---

1/2 49.91 24.5 0.052454 17.173 3.05 -0.5974

-2.6306

2.06

0.99

-0.9478 2.50

2/3 26.27 23.5 0.054264 18.08 3.14 -0.1746

-0.7471

-2.7639

28.1

6.39

1.68

-0.1921

-1.1617

29.55

5.74

Time Domain Estimation Results

Test at 55% NH

Test A2 HP Shaft

Order Cost Fn. Zeros σz

(%)

Poles σp

(%)

0/1 1.3140 --- --- 0.9878 0.0019

1/2 0.3173 0.9119 0.0062 0.9867

0.9232

0.0013

0.0029

2/3 0.2861 0.9842

0.8563

0.0071

0.0161

0.9859+0.0028i

0.9859-0.0028i

0.8738

0+0.0001i

0.3012

0.0001


Test A2 LP Shaft


(%)

Poles σp

(%)

0/1 0.7115 --- --- 0.9869 0.0019

1/2 0.2023 0.9056 0.0068 0.9859

0.9174

0.0013

0.0031

2/3 0.2031 0.9892

0.6572

0.0084

0.0496

0.9879+0.0019i

0.9879-0.0019i

0.7887

0.0+0.008i

-23-0.04i

0.0092

Test C1 HP Shaft


(%)

Poles σp

(%)

0/1 1.2176 --- --- 0.9913 0.0078

1/2 0.0361 0.7304 0.0517 0.9897

0.7969

0.0027

0.0044

2/3 0.0070 0.9538

-0.5040

0.0806

-5.6035

0.9891

0.9576

0.5860

0.0430

0.1324

0.1281

Test C1 LP Shaft


(%)

Poles σp

(%)

0/1 0.2420 --- --- 0.9897 0.0069

1/2 0.0085 0.6063 0.1733 0.9884

0.6988

0.0019

0.0863


Test A3 HP Shaft


(%)

Poles σp

(%)

0/1 3.9872 --- --- 0.9897 0.0041

1/2 0.2085 0.9144 0.0064 0.9872

0.9305

0.0026

0.0036

2/3 0.1226 0.9667

0.4594

0.042

0.1622

0.9857

0.9728

0.7120

0.0197

0.0618

0.0498

Test A3 LP Shaft


(%)

Poles σp

(%)

0/1 2.1158 --- --- 0.9889 0.0043

1/2 0.1521 0.9014 0.0105 0.9865

0.9195

0.0034

0.0049

2/3 0.0932 0.9714

0.3877

0.0242

0.3704

0.9835

0.9782

0.7024

0.0308

0.0511

0.0278

Test at 65% NH

Test A4 HP Shaft


(%)

Poles σp

(%)

0/1 2.8148 --- --- 0.9815 0.0057

1/2 0.4257 0.1181 3.0598 0.9797

0.5156

0.0024

0.0300


Test A4 LP Shaft


(%)

Poles σp

(%)

0/1 3.9111 --- --- 0.9735 0.0064

1/2 0.3331 0.9249 0.0042 0.9786

0.9057

0.0031

0.0072

2/3 0.1686 0.9724

0.8501

0.0525

0.0465

0.9826

0.9624

0.8288

0.0159

0.0439

0.0684

Test B10 HP Shaft


(%)

Poles σp

(%)

0/1 0.3502 --- --- 0.9814 0.0144

1/2 0.3236 0.7713 0.4053 0.9801

0.7929

0.0218

0.23989

Test B10 LP Shaft


(%)

Poles σp

(%)

0/1 0.7775 --- --- 0.9741 0.0126

1/2 0.3637 0.8761 0.223

0

0.9771

0.9603

0.0239

0.2129


Test at 75% NH

Test A6 HP Shaft


(%)

Poles σp

(%)

0/1 1.2424 --- --- 0.9735 0.0025

1/2 0.4895 0.9487 0.0057 0.9754

0.9408

0.0033

0.0028

2/3 0.1930 0.9828

0.0929

0.0169

0.7309

0.9851

0.9685

0.4566

0.0139

0.0056

0.0218

Test A6 LP Shaft


(%)

Poles σp

(%)

0/1 9.6100 --- --- 0.9605 0.0132

1/2 0.3983 0.9397 0.0086 0.9744

0.8931

0.0032

0.0045

2/3 0.1379 0.9830

0.9095

0.0051

0.0115

0.9861

0.9636

0.8621

0.0058

0.0056

0.0046


Test A8 HP Shaft


(%)

Poles σp

(%)

0/1 1.0289 --- --- 0.9720 0.0030

1/2 0.6499 0.9407 0.0079 0.9753

0.9334

0.0105

0.0181

2/3 0.2064 0.9804

-0.1683

0.1567

-1.5400

0.9829

0.9690

0.3955

0.1109

0.0576

0.2307

Test A8 LP Shaft

Order Cost

Fn.

Zeros σz

(%)

Poles σp

(%)

0/1 4.9941 --- --- 0.9600 0.0092

1/2 0.3877 0.9276 0.0067 0.9723

0.8823

.00067

0.0167

2/3 0.2473 0.9438

-0.4369

0.0328

-3.0081

0.9747

0.9057

0.3215

0.0059

0.0479

0.1612

3/4 0.2359 0.96+0.1

7i

0.96-

0.17i

0.9404

0+0i

-18-3i

0.0243

0.9596+0.1725i

0.9596-0.1725i

0.9744

0.8968

0+0i

0.7+0.1i

0.0099

0.0626


Test A7 HP Shaft


(%)

Poles σp

(%)

0/1 2.2056 --- --- 0.9719 0.0031

1/2 1.4059 0.9158 0.0099 0.9740

0.9084

0.0042

0.0092

2/3 0.2566 -1.0898

0.9911

-0.6909

0.0082

0.9919

0.9711

0.1751

0.0076

0.0034

0.0236

3/4

Test A7 LP Shaft


(%)

Poles σp

(%)

0/1 8.2479 --- --- 0.9604 0.0088

1/2 0.6078 0.9310 0.0073 0.9728

0.8863

0.0054

0.0066

2/3 0.4128 0.9806

0.8912

0.0128

0.0150

0.9850

0.9582

0.8501

0.0118

0.0145

0.0159


Test B11 HP Shaft


(%)

Poles σp

(%)

0/1 1.6142 --- --- 0.9758 0.0157

1/2 0.2278 -231.29 -86.32 0.9432

-0.6566

0.0035

-0.7691

2/3 0.1809 -1.1885

0.8780

-6.0024

0.8292

0.9737

0.8733

0.1134

0.0369

0.7699

2.1677

Test B11 LP Shaft


(%)

Poles σp

(%)

0/1 6.0114 --- --- 0.9509 0.0320

1/2 0.9448 0.8844 0.0884 0.9667

0.8289

0.0286

0.0563

2/3 0.1454 -1.3392

0.9361

-1.6577

0.4007

0.9734

0.8946

0.1491

0.1162

0.3605

1.1291


Test at 80% NH

Test C12 HP Shaft


(%)

Poles σp

(%)

0/1 1.8760 --- --- 0.9612 0.0046

1/2 0.8436 0.8794 0.0079 0.9631

0.8524

0.0047

0.0119

2/3 0.2587 0.9745

0.4838

0.0038

0.1100

0.9790

0.9527

0.3010

0.0059

0.0076

0.0523

Test C12 LP Shaft


(%)

Poles σp

(%)

0/1 9.2170 --- --- 0.9303 0.0171

1/2 0.8549 0.9373 0.0056 0.9641

0.8553

0.0068

0.0166

2/3 0.2974 0.9773

0.8710

0.0330

0.0262

0.9821

0.9312

0.8012

0.0252

0.0211

0.0550


Test at 85% NH

Test A13 HP Shaft


(%)

Poles σp

(%)

0/1 1.7490 --- --- 0.9657 0.0044

1/2 0.7866 0.8793 0.0139 0.9669

0.8555

0.0046

0.0051

2/3 0.1749 0.9815

0.6188

0.0294

0.0660

0.9842

0.9596

0.5253

0.0233

0.0100

0.0457

Test 13 LP Shaft


(%)

Poles σp

(%)

0/1 9.4711 --- --- 0.9403 0.0159

1/2 0.6299 0.9348 0.0080 0.9669

0.8614

0.0063

0.0068

2/3 0.1602 0.9777

0.8862

0.0330

0.0225

0.9829

0.9426

0.8203

0.0228

0.0211

0.0434


Test B12 HP Shaft


(%)

Poles σp

(%)

0/1 2.7365 --- --- 0.9561 0.0202

1/2 0.7123 0.6248 0.1791 0.9612

0.5006

0.0091

0.0695

2/3 0.3326 -2.4395

0.9289

-0.8562

0.1076

0.9655

0.9171

-0.2241

0.0213

0.0916

-0.1343

Test B12 LP Shaft


(%)

Poles σp

(%)

0/1 4.2329 --- --- 0.9320 0.0319

1/2 0.9442 0.8930 0.1800 0.9547

0.8047

0.0444

0.2220

2/3 0.1663 0.9500

0.3364

0.1598

1.6262

0.9700

0.8842

0.0682

0.1004

0.0948

5.2262


Test at 90% NH

Test 11A HP Shaft


(%)

Poles σp

(%)

0/1 2.6982 --- --- 0.9597 0.0056

1/2 0.5422 0.9006 0.0785 0.9663

0.8349

0.0056

0.1571

1/3 0.0947 0.9818

0.7250

0.0125

0.0787

0.9838

0.9569

-0.4427

0.0098

0.0067

-1.8034

Test 11A LP Shaft


(%)

Poles σp

(%)

0/1 10.012 --- --- 0.9334 0.0260

1/2 0.5892 0.9457 0.0352 0.9691

0.8205

0.0123

0.1135

1/3 0.0785 0.9788

0.7982

0.0098

0.0589

0.9829

0.9314

-0.9723

0.0066

0.0160

-0.8001


Test 12A HP Shaft


(%)

Poles σp

(%)

0/1 3.5244 --- --- 0.9590 0.0060

1/2 0.8959 0.8002 0.1078 0.9633

0.6216

0.0042

0.3540

1/3 0.2401 0.9843

0.697

0.0091

0.0790

0.9859

0.9579

-0.9905

0.0077

0.0057

-0.3915

Test 12A LP Shaft


(%)

Poles σp

(%)

0/1 14.6488 --- --- 0.9261 0.0284

1/2 1.2841 0.9258 0.0491 0.9632

0.7831

0.0145

0.1452

Test 19C HP Shaft


(%)

Poles σp

(%)

0/1 0.6431 --- --- 0.9679 0.0056

1/2 0.2768 0.5602 1.1787 0.9693

-0.4733

0.0044

13.701

Test 19C LP Shaft


(%)

Poles σp

(%)

0/1 3.0872 --- --- 0.9357 0.0303

1/2 0.6668 0.9137 0.1279 0.9639

0.7936

0.0278

0.2822

Appendix 3 – Nonlinear Gas Turbine Estimation Results 338

Appendix 3 – Nonlinear Gas Turbine Estimation Results

________________________________________________

Table 3A-1

STATISTICAL CRITERIA COMPUTED FOR A VALIDATION DATA

SET AT 55% NH, nu=ny=2 l=2, HP Shaft

Model size NV AIC

×104

FPE

×104

BIC

×104

LILC

×104

1 *** *** *** *** ***

2 3.9105 0.5459 0.5455 0.5468 0.5460

3 76.712 1.7364 1.7364 1.7376 1.7364

4 0.1273 -0.8224 -0.8232 -0.8207 -0.8223

5 0.0515 -1.1836 -1.1846 -1.1815 -1.1835

6 0.0243 -1.4827 -1.4839 -1.4801 -1.4825

7 0.0341 -1.3473 -1.3487 -1.3443 -1.3471

8 0.0174 -1.6146 -1.6162 -1.6112 -1.6144

9 0.0180 -1.6008 -1.6026 -1.5970 -1.6006

10 0.0203 -1.5537 -1.5557 -1.5494 -1.5535

11 0.0204 -1.5507 -1.5529 -1.5460 -1.5505

12 0.0215 -1.5292 -1.5316 -1.5241 -1.5290

13 0.0202 -1.5547 -1.5573 -1.5491 -1.5544

14 0.0296 -1.4011 -1.4039 -1.3951 -1.4008

15 0.0298 -1.3974 -1.4004 -1.3909 -1.3970


Table 3A-2



Model size NV AIC

×104

FPE

×104

BIC

×104

LILC

×104

1 *** *** *** *** ***

2 9.9842 0.9207 0.9234 0.9216 0.9207

3 37.790 1.4533 1.4527 1.4546 1.4534

4 0.0270 -1.4412 -1.4420 -1.4394 -1.4411

5 0.0170 -1.6248 -1.6258 -1.6226 -1.6246

6 0.0311 -1.3850 -1.3862 -1.3824 -1.3848

7 0.0601 -1.1209 -1.1223 -1.1179 -1.1207

8 0.0272 -1.4376 -1.4392 -1.4342 -1.4374

9 0.0281 -1.4233 -1.4251 -1.4194 -1.4231

10 0.0293 -1.4072 -1.4092 -1.4029 -1.4070

11 0.0284 -1.4181 -1.4203 -1.4134 -1.4179

12 0.0300 -1.3960 -1.3984 -1.3908 -1.3957

13 0.0286 -1.4157 -1.4183 -1.4101 -1.4154

14 0.0496 -1.1951 -1.1979 -1.1891 -1.1947

15 0.0497 -1.1933 -1.1963 -1.1869 -1.1930


Table 3A-3



Model size NV AIC

×104

FPE

×104

BIC

×104

LILC

×104

1 *** *** *** *** ***

2 15.610 1.0994 1.0990 1.1003 1.0995

3 1.6026 0.1897 0.1891 0.1910 0.1898

4 0.0158 -1.6560 -1.6568 -1.6543 -1.6559

5 0.0122 -1.7574 -1.7584 -1.7552 -1.7573

6 0.0087 -1.8917 -1.8929 -1.8891 -1.8916

7 0.0158 -1.6554 -1.6568 -1.6524 -1.6552

8 0.0181 -1.6015 -1.6031 -1.5981 -1.6013

9 0.0180 -1.6022 -1.6040 -1.5983 -1.6020

10 0.0153 -1.6645 -1.6665 -1.6603 -1.6643

11 0.0156 -1.6585 -1.6607 -1.6538 -1.6582

12 0.0154 -1.6615 -1.6639 -1.6564 -1.6613

13 0.0159 -1.6481 -1.6507 -1.6425 -1.6478

14 0.0162 -1.6414 -1.6442 -1.6354 -1.6411

15 0.0159 -1.6480 -1.6510 -1.6415 -1.6476


Table 3A-4



Model size NV AIC

×104

FPE

×104

BIC

×104

LILC

×104

1 *** *** *** *** ***

2 19.553 1.1894 1.1890 1.1903 1.1895

3 127.40 1.9392 1.9386 1.9405 1.9392

4 0.0156 -1.6602 -1.6610 -1.6585 -1.6601

5 0.0096 -1.8529 -1.8539 -1.8508 -1.8528

6 0.0124 -1.7515 -1.7527 -1.7490 -1.7514

7 0.0187 -1.5878 -1.5892 -1.5848 -1.5876

8 0.0168 -1.6300 -1.6316 -1.6265 -1.6298

9 0.0177 -1.6084 -1.6102 -1.6045 -1.6082

10 0.0168 -1.6282 -1.6302 -1.6239 -1.6280

11 0.0163 -1.6403 -1.6425 -1.6356 -1.6400

12 0.0165 -1.6341 -1.6365 -1.6290 -1.6339

13 0.0164 -1.6364 -1.6390 -1.6308 -1.6361

14 0.0177 -1.6072 -1.6100 -1.6012 -1.6069

15 0.0177 -1.6058 -1.6088 -1.5994 -1.6055


Table 3A-5


SET AT 55% NH, nu=ny=2 l=2 (STANDARDISED DATA) , HP Shaft

Model size NV AIC

×104

FPE

×104

BIC

×104

LILC

×104

1 *** *** *** *** ***

2 77.018 1.7380 1.7376 1.7388 1.7380

3 76.712 1.7364 1.7358 1.7376 1.7364

4 1.1231 0.0480 0.0472 0.0497 0.0481

5 0.0303 -1.3954 -1.3964 -1.3932 -1.3952

6 0.0341 -1.3482 -1.3494 -1.3457 -1.3481

7 0.0183 -1.5949 -1.5963 -1.5919 -1.5947

8 0.0188 -1.5847 -1.5863 -1.5813 -1.5846

9 0.0188 -1.5847 -1.5865 -1.5808 -1.5845

10 0.0191 -1.5765 -1.5785 -1.5722 -1.5763

11 0.0211 -1.5368 -1.5390 -1.5320 -1.5365

12 0.0199 -1.5607 -1.5631 -1.5556 -1.5605

13 0.0203 -1.5522 -1.5548 -1.5466 -1.5519

14 0.0296 -1.4011 -1.4039 -1.3951 -1.4008

15 0.0298 -1.3974 -1.4004 -1.3909 -1.3909


Table 3A-6


SET AT 65% NH, nu=ny=2 l=2 (STANDARDISED DATA) , HP Shaft

Model size NV AIC

×104

FPE

×104

BIC

×104

LILC

×104

1 *** *** *** *** ***

2 38.100 1.4565 1.4561 1.4574 1.4566

3 37.790 1.4533 1.4527 1.4546 1.4534

4 0.2628 -0.5326 -0.5334 -0.5308 -0.5325

5 0.0750 -1.0331 -1.0341 -1.0310 -1.0330

6 0.0630 -1.1028 -1.1040 -1.1002 -1.1027

7 0.0403 -1.2809 -1.2823 -1.2779 -1.2807

8 0.0412 -1.2718 -1.2734 -1.2683 -1.2716

9 0.0259 -1.4564 -1.4582 -1.4526 -1.4562

10 0.0276 -1.4311 -1.4331 -1.4268 -1.4309

11 0.0306 -1.3894 -1.3916 -1.3847 -1.3891

12 0.0291 -1.4091 -1.4115 -1.4040 -1.4088

13 0.0322 -1.3683 -1.3709 -1.3627 -1.3680

14 0.0496 -1.1951 -1.1979 -1.1891 -1.1947

15 0.0497 -1.1933 -1.1963 -1.1869 -1.1930


Table 3A-7


SET AT 75% NH, nu=ny=2 l=2 (STANDARDISED DATA), HP Shaft

Model size NV AIC

×104

FPE

×104

BIC

×104

LILC

×104

1 *** *** *** *** ***

2 1.5547 0.1772 0.1768 0.1781 0.1773

3 1.6026 0.1897 0.1891 0.1910 0.1898

4 0.4462 -0.3210 -0.3218 -0.3192 -0.3209

5 0.0151 -1.6744 -1.6754 -1.6722 -1.6743

6 0.0165 -1.6373 -1.6385 -1.6347 -1.6372

7 0.0244 -1.4804 -1.4818 -1.4774 -1.4803

8 0.0242 -1.4832 -1.4848 -1.4797 -1.4830

9 0.0159 -1.6500 -1.6518 -1.6461 -1.6497

10 0.0161 -1.6465 -1.6485 -1.6422 -1.6462

11 0.0152 -1.6670 -1.6692 -1.6623 -1.6667

12 0.0157 -1.6537 -1.6561 -1.6485 -1.6534

13 0.0154 -1.6621 -1.6647 -1.6565 -1.6618

14 0.0162 -1.6414 -1.6442 -1.6354 -1.6411

15 0.0159 -1.6480 -1.6510 -1.6415 -1.6476


Table 3A-8


SET AT 85% NH, nu=ny=2 l=2 (STANDARDISED DATA), HP Shaft

Model size NV AIC

×104

FPE

×104

BIC

×104

LILC

×104

1 *** *** *** *** ***

2 126.47 1.9363 1.9359 1.9372 1.9364

3 127.40 1.9392 1.9386 1.9405 1.9392

4 0.2970 -0.4836 -0.4844 -0.4819 -0.4835

5 0.0218 -1.5266 -1.5276 -1.5244 -1.5265

6 0.0191 -1.5782 -1.5794 -1.5756 -1.5781

7 0.0203 -1.5543 -1.5557 -1.5513 -1.5541

8 0.0213 -1.5346 -1.5362 -1.5312 -1.5344

9 0.0165 -1.6351 -1.6369 -1.6312 -1.6349

10 0.0167 -1.6313 -1.6333 -1.6270 -1.6311

11 0.0170 -1.6230 -1.6252 -1.6183 -1.6228

12 0.0169 -1.6253 -1.6277 -1.6202 -1.6251

13 0.0179 -1.6012 -1.6038 -1.5956 -1.6009

14 0.0177 -1.6072 -1.6100 -1.6012 -1.6069

15 0.0177 -1.6058 -1.6088 -1.5994 -1.6055


40 60 8050

54

58

Time (s)

Sha

ft S

peed

(%

NH

)

(a)

40 60 80

62

64

66

Time (s)

Sha

ft S

peed

(%

NH

)

(b)

40 60 8072

74

76

78

Time (s)

Sha

ft S

peed

(%

NH

)

(c)

40 60 80

82

85

88

Time (s)

Sha

ft S

peed

(%

NH

)

(d)

100 120 140 18056

64

72

Time (s)

Sha

ft S

peed

(%

NH

)

(e)

80 120 160 20065

75

85

Time (s)

Sha

ft S

peed

(%

NH

)

(f)

Figure 3A-1. Outputs of validation data sets, HP shaft. Measured output (solid), Model

1 output (dashed). (a) multisine (1) test at 55% NH, (b) IRMLBS test at 65% NH, (c)

multisine (1) test at 75% NH, , (d) IRMLBS test at 85% NH, , (e) three-level periodic

test at 58-70% NH, (f) triangular wave + IRMLBS test at 65-85% NH.

40 60 8050

54

58

Time (s)

Sha

ft S

peed

(%

NH

)

(a)

40 60 80

62

64

66

Time (s)

Sha

ft S

peed

(%

NH

)

(b)

40 60 8072

74

76

78

Time (s)

Sha

ft S

peed

(%

NH

)

(c)

40 60 80

82

85

88

Time (s)

Sha

ft S

peed

(%

NH

)

(d)

100 120 140 18056

64

72

Time (s)

Sha

ft S

peed

(%

NH

)

(e)

80 120 160 20065

75

85

Time (s)

Sha

ft S

peed

(%

NH

)

(f)






40 60 8050

54

58

Time (s)

Sha

ft S

peed

(%

NH

)

(a)

40 60 80

62

64

66

Time (s)

Sha

ft S

peed

(%

NH

)

(b)

40 60 8072

74

76

78

Time (s)

Sha

ft S

peed

(%

NH

)

(c)

40 60 80

82

85

88

Time (s)

Sha

ft S

peed

(%

NH

)

(d)

100 120 140 18056

64

72

Time (s)

Sha

ft S

peed

(%

NH

)

(e)

80 120 160 20065

75

85

Time (s)

Sha

ft S

peed

(%

NH

)

(f)





40 60 80

25

26

27

28

Time (s)

Sha

ft S

peed

(%

NH

)

(a)

40 60 80

32

34

36

Time (s)

Sha

ft S

peed

(%

NH

)

(b)

40 60 8040

43

46

Time (s)

Sha

ft S

peed

(%

NH

)

(c)

40 60 80

48

50

52

Time (s)

Sha

ft S

peed

(%

NH

)

(d)

100 120 140 18028

34

40

Time (s)

Sha

ft S

peed

(%

NH

)

(e)

80 120 160 200

35

40

45

50

Time (s)

Sha

ft S

peed

(%

NH

)

(f)

Figure 3A-4. Outputs of validation data sets, LP shaft. Measured output (solid), Model


multisine (1) test at 75% NH, (d) IRMLBS test at 85% NH, (e) three-level periodic test at

58-70% NH, (f) triangular wave + IRMLBS test at 65-85% NH.


40 60 80

25

26

27

28

Time (s)

Sha

ft S

peed

(%

NH

)

(a)

40 60 80

32

34

36

Time (s)

Sha

ft S

peed

(%

NH

)

(b)

40 60 8040

43

46

Time (s)

Sha

ft S

peed

(%

NH

)

(c)

40 60 80

48

50

52

Time (s)

Sha

ft S

peed

(%

NH

)

(d)

100 120 140 18028

34

40

Time (s)

Sha

ft S

peed

(%

NH

)

(e)

80 120 160 200

35

40

45

50

Time (s)

Sha

ft S

peed

(%

NH

)

(f)





40 60 80

25

26

27

28

Time (s)

Sha

ft S

peed

(%

NH

)

(a)

40 60 80

32

34

36

Time (s)

Sha

ft S

peed

(%

NH

)

(b)

40 60 8040

43

46

Time (s)

Sha

ft S

peed

(%

NH

)

(c)

40 60 80

48

50

52

Time (s)

Sha

ft S

peed

(%

NH

)

(d)

100 120 140 18028

34

40

Time (s)

Sha

ft S

peed

(%

NH

)

(e)

80 120 160 200

35

40

45

50

Time (s)

Sha

ft S

peed

(%

NH

)

(f)






40 60 80

25

26

27

28

Time (s)

Sha

ft S

peed

(%

NH

)

(a)

40 60 80

32

34

36

Time (s)

Sha

ft S

peed

(%

NH

)

(b)

40 60 8040

43

46

Time (s)

Sha

ft S

peed

(%

NH

)

(c)

40 60 80

48

50

52

Time (s)

Sha

ft S

peed

(%

NH

)

(d)

100 120 140 18028

34

40

Time (s)

Sha

ft S

peed

(%

NH

)

(e)

80 120 160 200

35

40

45

50

Time (s)

Sha

ft S

peed

(%

NH

)

(f)



multisine (1) test at 75% NH, (d) IRMLBS test at 85% NH, (e) three-level periodic test

at 58-70% NH, (f) triangular wave + IRMLBS test at 65-85% NH.

Documents

Appendix 1 – Terms and Concepts in Model Estimationagrino.org/nchiras/research/pubs/thesis/Appendices.pdfAppendix 1 – Terms and Concepts in Model Estimation 311 The variances of